In general, oil is not healthy when burnt. Burnt oil has many negative health effects.
As far as oils go, extra virgin olive oil has the very desirable property of a high smoke point, which means it is less likely to burn when cooked, and makes it an option for frying. That said, more refined oils like olive pomace oil have an even higher smoke point and lighter taste making them ideal for frying. (Ref.)
Keep also in mind that the quality of the oil makes the smoke point vary significantly.
That said, as far as oils go, olive pomace oil, virgin olive oil and extra virgin olive oil are all "healthy" oils given low levels of saturated and polyunsaturated fats.
Regarding cancer, extra virgin olive oil is considered a potential anticarcinogenic, so quite the opposite of the claim.
Compared to other vegetable oils, the presence of several phenolic antioxidants in olive oil is believed to prevent the occurrence of a variety of pathological processes, such as cancer.
There have been numerous evidences supporting the relationship between olive oil and cancer, with most of the attention being directed toward its fat and phenolic content. [...] It was observed that upon combination treatment, anti-proliferation effects and apoptosis induction were augmented.
Many of these studies are more generally about the Mediterranean diet, where olive oil of all kinds is commonly used in cooking.